# Expected correlation coefficient given different ranges of values of one variable

I want to do a study looking at Pearson's correlation between two variables, let's call them x and y. I know the values of y for a number of samples. I believe that a higher correlation coefficient is more likely when looking at samples with a larger range of values of the variables of interest than if restricted to a smaller range. If I have pilot data from which I have calculated a correlation coefficient for a given range of y values e.g. from 1 to 5, i.e. a 5 fold change between minimum and maximum, is there any way of estimating expected correlation coefficients using samples with different ranges, e.g. samples with y values ranging from 1 to 10 i.e. a 10 fold change between minimum and maximum, or samples with y values ranging from 1 to 2 i.e. only a 2 fold change between minimum and maximum?

Assuming you know the standard deviations rather than the ranges there is a formula you can use. If we let $\Sigma$ be the unrestricted standard deviation, $\sigma$ the restricted, $\rho$ a correlation and suppose that restriction is made on $x$ then the correlation corrected for range restriction is

$$\frac{\rho_{xy}\frac{\Sigma_x}{\sigma_x}}{\sqrt{1-\rho_{xy}^2+\rho_{xy}^2\frac{\Sigma_x^2}{\sigma_x^2}}}$$

I have copied the formula from a paper by Stauffer and Mendoza available from Psychometrika here in volume 66 (2001) 63-68. You might want to double check my typing before using it but I believe it to be OK.

You may want to give a read to the "correlation fallacies" section in 7.2.2 of the copulas and dependence slides in this site: http://qrmtutorial.org/slides. Specifically to the part "given F1 and F2, any rho in [-1,1] is attainable".

They provide a very interesting example in which they take two lognormals, and derive correlation bounds for them given different sigma values for one of the lognormals. For obtaining these bounds, Hoeffding copula bounds are used (see here too).

So it doesn't exactly give you an expected correlation coefficient, but it gives you bounds over the correlations between two distributions.

You could then verify if the correlations you get for different realizations in different ranges of the distributions satisfy those bounds.